Tail Equivalent Linearization Method Research Articles

Application of the Tail Equivalent Linearization Method to wave bending moment and comparison with experimental data

This paper estimates extreme midship wave bending moment of a ship in a specific sea state using First Order Reliability Method (FORM) and the Tail Equivalent Linearization Method (TELM). FORM and TELM are applied to the random vibration analysis of the nonlinear bending moment and results are compared and verified against experimental data obtained from model tests as well as numerical simulations. A case study of an LNG tanker is presented, demonstrating that the proposed methodology yields reasonable results and that the TELM approach provides a fast and computationally efficient way of analyzing nonlinear vertical bending moment of ships in stationary sea states.

Nonlinear stochastic dynamic analysis by evolutionary tail-equivalent linearization method

This study introduces the evolutionary tail-equivalent linearization method (ETELM) for nonlinear stochastic dynamic analysis. The method builds on the recently developed tail-equivalent linearization method (TELM) and it is designed for the class of evolutionary processes. The original TELM employs a tail-equivalent linear system (TELS) by equating the tail probability of a linear system response for a specified threshold to the first-order approximation of the tail probability of the nonlinear system response. For stationary problems, the TELS is time-independent and only one linear system needs to be defined for the specified threshold. However, for a transient input, the TELS is time dependent and an evolutionary tail-equivalent linear system (ETELS) must be defined to study the entire transient response. Algorithms are developed to determine a discrete-time ETELS based on a sequence of linearization points, and a continuous-time approximation based on Priestley’s evolutionary theory. The linearized evolutionary system is used to compute the response statistics of interest, including the first-passage probability, in first-order approximation. Numerical examples demonstrate the accuracy and limitations of the proposed method.

Random vibration of nonlinear structures with stiffness and strength deterioration by modified tail equivalent linearization method

In this research the tail equivalent linearization method (TELM) has been extended to study structures with degrading materials. The responses of such structures to excitations are non-stationary, even if the excitations are stationary. Non-stationary behavior of the system cannot be considered by conventional TELM. Applying the conventional TELM, the only distinction in the design point excitation for two stationary excitations with different durations is in the addition of a zero value part at the beginning of the design point of the longer excitation. This means that the failure probability is the same for the non-stationary systems under excitations with different durations. Therefore, this solution cannot be correct. In this study, in using TELM for systems with degrading materials, hysteretic energy is replaced by average hysteretic energy, calculated by averaging the obtained hysteretic energy of the structure subjected to a few random sample load realization. In this way, the degradation parameters under design point coincide with those under sample load realizations. Since the average of the hysteretic energy is converges very fast, the modified TELM only requires about tens to hundreds solutions of the response in addition to the ordinary calculations of conventional TELM.

The Non-gradient-based Reliability Method in Equivalent Linear Systems for Nonlinear Random Vibration

A conventional approach for nonlinear random vibration analysis is using equivalent linearization method. Tail-Equivalent Linearization Method (TELM) is one the best proposed methods in the recent decade for determination of equivalent linear model. In TELM, the design point is obtained using first-order reliability method. In the current research, a non-gradient-based method is applied for determination of the design point. One of the main advantages of this method is non-application of limit-state function gradient for calculation of the design point. In the implemented method, n arbitrary points in n-dimension standard normal space are selected and limit-state function in these points is estimated. Then, these points converge to the design point using a convergent algorithm. Since many random variables are produced in TELM for discretization of seismic excitation, iterative algorithms for determination of design point numerical instability would be encountered. By modification of step length for each iteration and application of a magnification coefficient for each step, an appropriate non-gradient method is proposed for analysis of the problems with many random variables. The efficiency of this method was investigated by solving numerical examples. Moreover, the convergence of this method for finding the design point was presented. It was also indicated that results obtained using this method are in good agreement with results obtained by gradient methods.

Simulation of Stochastic Processes by Sinc Basis Functions and Application in TELM Analysis

This study serves three purposes: (1) to review a synthesis formula for simulation of band-limited stochastic processes based on the sinc expansion; (2) to implement this synthesis formula in the tail-equivalent linearization method (TELM); and (3) to demonstrate increased computational efficiency when the sinc expansion is implemented in this context. The proposed representation enables the reduction and control of the number of random variables used in the simulation of band-limited stochastic processes. This is of great importance for gradient-based reliability methods, including TELM, for which the computational cost is proportional to the total number of random variables. A direct application of the representation is used in TELM analysis. Examples of single-degree-of-freedom and multiple-degrees-of-freedom nonlinear systems subjected to Gaussian band-limited white noise simulated by use of sinc expansion are presented. The accuracy and efficiency of the representation are compared with those of the current time-domain discretization method used in TELM. The analysis concludes by shedding light on the specific cases for which the introduced reduction technique is beneficial.

Bivariate Gaussian mixture–based equivalent linearization method for stochastic seismic analysis of nonlinear structures

SummaryTo address challenges in stochastic seismic analysis of nonlinear structures, this paper further develops a recently proposed Gaussian mixture–based equivalent linearization method (GM‐ELM). The GM‐ELM uses a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. Using properties of the Gaussian mixture model, GM‐ELM can decompose the non‐Gaussian response of a nonlinear system into multiple Gaussian responses of linear single–degree of freedom oscillators. With the set of the equivalent linear systems identified by GM‐ELM, response statistics as crossing rate and first‐passage probability can be computed conveniently using theories of linear random vibration analysis. However, the original version of GM‐ELM may lead to an inaccurate estimate because of the heuristic parameters of the linear system introduced to supplement insufficient information. To overcome this limitation and define unique equivalent linear systems, this paper proposes a further developed version of GM‐ELM, which uses a mixture of bivariate Gaussian densities instead of univariate models. Moreover, to facilitate the use of elastic response spectra for estimating the mean peak responses of a nonlinear structure, a new response spectrum combination rule is proposed for GM‐ELM. Two numerical examples of hysteretic structural systems are presented in this paper to illustrate the application of the bivariate GM‐ELM to nonlinear stochastic seismic analysis. The analysis results obtained by the bivariate GM‐ELM are compared with those obtained by the univariate GM‐ELM, the conventional equivalent linearization method, the tail equivalent linearization method, and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/yisangri/GitHub‐bGM‐ELM‐code.git

A new approach for finding the design point of nonlinear systems under random excitation

In order to perform nonlinear stochastic dynamic analyses, the tail equivalent linearization method may be employed, which entails the adoption of the first-order reliability method. The major computational challenge posed in this method is related to finding the design point excitation for the threshold corresponding to the failure of the nonlinear system. Furthermore, finding the design point is also a sufficiently challenging problem in some importance sampling techniques. In the present paper, a new approach for finding the design point of nonlinear SDOF systems under white noise excitation is introduced, one that is identical to the mirror image excitation method for a nonlinear elastic system and yields superior results for a nonlinear hysteresis system compared to the results of mirror image excitation method. It is then extended to filtered white noise excitations and multi degree of freedom systems. Some numerical examples are given for confirming the efficiency and accuracy of the proposed method by comparing it to other methods.

Nonlinear Biaxial Structural Vibration under Bidirectional Random Excitation with Incident Angle θ by Tail-Equivalent Linearization Method

AbstractIn this research the tail-equivalent linearization method (TELM) has been applied to a structure with biaxial behavior of materials, using the biaxial Bouc-Wen material model. The modeled structure has been subjected to independent bidirectional excitation with the incident angle θ with major axes of structure. The direct differentiation method (DDM) has been developed for calculating the response and its derivatives for the first time for the biaxial Bouc-Wen material model, where the application of DDM is more difficult compared with uniaxial Bouc-Wen models due to its coupled constitutive law of material. The method is applied to a structure with a rigid diaphragm, supported by four different columns. The structure is subjected to bidirectional and modulated filtered white noise excitations. The cumulative probability distribution function (CDF), probability density function (PDF), average rate of crossing, and first passage probability of displacement response are calculated for a column in th...

Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions

Efficient estimate of the small first passage probabilities of nonlinear structures under random excitations is of great importance in structural reliability analysis. Some methods, such as the subset simulation method, tail equivalent linearization method, asymptotic sampling method, Monte Carlo simulation method based on the extreme value theory, etc., have been developed for estimating the probabilities, however, the efficiency of the estimate is still a challenging task. In the present study, a new method is developed to overcome the challenge. The method approximates the tails of the univariate extreme distributions of the responses by using the shifted generalized lognormal distributions, in which the model parameters are estimated by an efficient method called the extrapolation method. Based on the approximate tail distributions and covariances of the extreme responses, the tails of the multivariate extreme distributions of the nonlinear response are determined by using the Nataf model. Finally, from the relationship between the first passage probabilities and the extreme value distributions, the small first passage probabilities of interest can be estimated. The efficiency of the developed method is illustrated by the small first passage probability analyses of a single-degree-of-freedom system subjected to a stationary Gaussian process and two connected electric substation equipment items subjected to earthquake base motion.

Multicomponent Nonlinear Stochastic Dynamic Analysis by Tail-Equivalent Linearization

This paper extends the Tail-Equivalent Linearization Method (TELM) to the case of a nonlinear mechanical system subjected to multiple stochastic excitations. Following the original formulation, the method employs a discrete representation of the stochastic inputs and the first-order reliability method (FORM). Each component of the Gaussian excitation is expressed as a linear function of standard normal random variables. For a specified response threshold of the nonlinear system, the Tail-Equivalent Linear System (TELS) is defined in the standard normal space by matching the design points of the equivalent linear and nonlinear systems. This leads to the identification of the TELS in terms of a frequency-response function or, equivalently, an impulse-response function relative to each component of the input excitation. The method is demonstrated through its application to an asymmetric, one-story building with hysteretic behavior and subjected to bi-component ground motion. The degree of asymmetry is controlled by the eccentricity of the center of stiffness with respect to the center of mass. The correlation between the probability of failure and the degree of asymmetry is studied in detail. The statistics of the response for stationary excitation obtained by TELM are in close agreement with Monte Carlo simulation results.

Nonlinear random vibration using updated tail equivalent linearization method

Tail equivalent linearization method is based on first order reliability method, which obtains an equivalent linear system for the considered nonlinear problem with equal tail probability related to a specified threshold and time. This method has been applied only to nonlinear non-degrading single- and multi-degrees of freedom shear beam two-dimensional models and three-dimensional one-story rigid diaphragm supported by frames with in-plane uni-axial stiffness which is subjected to independent random excitation along the structural axes. To use TELM for more practical problems it is required to extend this method to cover more realistic material and excitation characteristics. In this paper, some of these developments have been presented. Application of TELM for bi-directional excitation with bi-axial material subjected to different incidence angles of excitation and using TELM for degrading material which has been presented in the previous works of the authors have been reviewed briefly. In addition a new method for defining rotational dependent component of earthquake excitation in terms of independent translational components in the standard normal random variable space is proposed, and TELM has been used for this kind of excitation. Three examples related to these extensions have been presented; the comparison of the TELM results with Mote-Carlo simulation results shows good agreement.

A gradient-free method for determining the design point in nonlinear stochastic dynamic analysis

A gradient-free method is developed for finding the design point in nonlinear stochastic dynamic analysis, where the input excitation is discretized into a large number of random variables. This point defines the realization of the excitation that is most likely to give rise to a specific response threshold at a given time. The design point is the essential information in the recently developed tail-equivalent linearization method. The proposed approach employs a variant of the model correction factor method developed by O. Ditlevsen, which is further improved by the use of a novel response surface technique. Example applications to single- and multi-degree-of-freedom hysteretic systems demonstrate the efficiency and accuracy of the method.

Tail-Equivalent Linearization Method in frequency domain and application to marine structures

An extension of the Tail-Equivalent Linearization Method (TELM) to the frequency domain is presented. The extension defines the Tail-Equivalent Linear System in terms of its frequency-response function. This function is obtained by matching the design point of the nonlinear response with that of the linearized response, thus guaranteeing the equivalence of the tail probability of the latter and the first-order approximation of the tail probability of the nonlinear response. The proposed approach is particularly suitable when the input and response processes are stationary, as is usually the case in the analysis of marine structures. When linear waves are considered, the Tail-Equivalent Linear System possesses a number of important properties, such as the capability to account for multi-support excitations and invariance with respect to scaling of the excitation. The latter property significantly enhances the computational efficiency of TELM for analysis with variable sea states. Additionally, the frequency-response function of the Tail-Equivalent Linear System offers insights into the geometry of random vibrations discretized in the frequency domain and into the physical nature of the response process. The proposed approach is applied to the analysis of point-in-time and first-passage statistics of the random sway displacement of a simplified jack-up rig model.

Tail-equivalent linearization method for nonlinear random vibration

A new, non-parametric linearization method for nonlinear random vibration analysis is developed. The method employs a discrete representation of the stochastic excitation and concepts from the first-order reliability method, FORM. For a specified response threshold of the nonlinear system, the equivalent linear system is defined by matching the “design points” of the linear and nonlinear responses in the space of the standard normal random variables obtained from the discretization of the excitation. Due to this definition, the tail probability of the linear system is equal to the first-order approximation of the tail probability of the nonlinear system, this property motivating the name Tail-Equivalent Linearization Method (TELM). It is shown that the equivalent linear system is uniquely determined in terms of its impulse response function in a non-parametric form from the knowledge of the design point. The paper examines the influences of various parameters on the tail-equivalent linear system, presents an algorithm for finding the needed sequence of design points, and describes methods for determining various statistics of the nonlinear response, such as the probability distribution, the mean level-crossing rate and the first-passage probability. Applications to single- and multi-degree-of-freedom, non-degrading hysteretic systems illustrate various features of the method, and comparisons with results obtained by Monte Carlo simulations and by the conventional equivalent linearization method (ELM) demonstrate the superior accuracy of TELM over ELM, particularly for high response thresholds.

On alternative approaches to linearization and Morison's equation for wave forces

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Wolfram Julian 1999On alternative approaches to linearization and Morison's equation for wave forcesProc. R. Soc. Lond. A.4552957–2974http://doi.org/10.1098/rspa.1999.0434SectionRestricted accessOn alternative approaches to linearization and Morison's equation for wave forces Julian Wolfram Julian Wolfram Department of Civil and shore Engineering, Heriot–Watt University, Edinburgh EH9 1UD, UK Google Scholar Find this author on PubMed Search for more papers by this author Julian Wolfram Julian Wolfram Department of Civil and shore Engineering, Heriot–Watt University, Edinburgh EH9 1UD, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 August 1999https://doi.org/10.1098/rspa.1999.0434AbstractThe limitations of the ‘equivalent linearization’ approach are discussed and an alternative approach is advocated based on the idea of ‘corresponding linear forms’ which give unbiased estimates for the output quantity required from an analysis. The application of the alternative approach is illustrated by developing corresponding linear forms for Morison's equation applicable to the estimation of extreme wave loading and the estimation of wave–induced fatigue loading. These corresponding linear forms differ only in a linearization factor and expressions for this are obtained for both cases. It is shown that the linearization factor varies not just with the nature of the output but also with the relative magnitudes of the drag and inertia terms in Morison's equation. The application of the alternative approach to the oscillatory motion of bodies is discussed briefly. 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